Abstract

Based on Riccati transformation and certain inequality technique,
some new oscillatory criteria are established for the solutions of a
class of sequential differential equations with fractional order defined in the modified Riemann-Liouville derivative. The
oscillatory criteria established are of new forms compared with the
existing results so far in the literature. For illustrating the
validity of the results established, we present some examples for
them.

1. Introduction

Recently, research for oscillation of various equations including differential equations, difference equations, and dynamic equations on time scales, has been a hot topic in the literature, and much effort has been done to establish new oscillation criteria for these equations so far (e.g., see [1–21] and the references therein). In these investigations, we notice that very little attention is paid to oscillation of fractional differential equations. Recent results in this direction only include Chen’s work [22, 23] and Zheng’s work [24].

In this paper, we are concerned with oscillation of a class of fractional differential equations as follows:
where denotes the modified Riemann-Liouville derivative [25] with respect to the variable , is the quotient of two odd positive numbers, the functions , denotes continuous derivative of order , and the function is continuous satisfying for some positive constant and for all .

The definition and some important properties for Jumarie’s modified Riemann-Liouville derivative of order are listed as follows (see also in [26–28]):

As usual, a solution of (1) is called oscillatory if it has arbitrarily large zeros; otherwise it is called nonoscillatory. Equation (1) is called oscillatory if all its solutions are oscillatory.

We organize this paper as follows. In Section 2, using Riccati transformation, inequality, and integration average technique, we establish some new oscillatory criteria for (1), while we present some examples for them in Section 3.

Lemma 1. Assume that is an eventually positive solution of (1), and
Then there exists a sufficiently large such that on and either on or .

Proof. Let , where . Then by use of (2) we obtain , and furthermore by use of the first equality in (4), we have
Similarly we have . So (1) can be transformed into the following form:
Since is an eventually positive solution of (1), then is an eventually positive solution of (9), and there exists such that on . Furthermore, we have
Then is strictly decreasing on , and thus is eventually of one sign. We claim on , where is sufficiently large. Otherwise, assume that there exists a sufficiently large such that on . Then is strictly decreasing on , and we have
By (5), we have . So there exists a sufficiently large with such that . Furthermore,
By (6) we deduce that , which contradicts the fact that is an eventually positive solution of (6). So on , and on . Thus is eventually of one sign. Now we assume that , for some sufficiently large . Since , furthermore we have . We claim that . Otherwise, assume that . Then on , and for by (9) we have
Substituting with in the inequality previously, an integration with respect to from to yields
which means
Substituting with in (15), an integration for (15) with respect to from to yields
that is,
Substituting with in (17), an integration for (17) with respect to from to yields
By (7), one can see , which causes a contradiction. So the proof is complete.

Lemma 2. Assume that is an eventually positive solution of (1) such that
on , where is sufficiently large. Then one has

Proof. By (10), we obtain that is strictly decreasing on . So
that is
which admits (20). On the other hand, we have
which can be rewritten as (21). So the proof is complete.

Lemma 3 (see [29, Theorem 41]). Assume that and are nonnegative real numbers. Then
for all .

Theorem 4. Let satisfying , and has continuous partial derivatives and on . Assume that (5)–(7) hold, and for any sufficiently large , there exist and and , and with satisfying
where . Then every solution of (1) is oscillatory or satisfies .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that on , where is sufficiently large. By Lemma 1 we have , where is sufficiently large and either on or . Now we assume that on . Define the generalized Riccati function
Then for , we have
By Lemma 2 and the definition of we get that
Using the following inequality (see [2, Equation ])
we obtain
A combination of (29) and (31) yields:
Let . Then , and (32) is transformed into the following form:
Select arbitrarily in with . Substituting with , multiplying both sides of (33) by , and integrating it with respect to from to for , we get that
Setting
by a combination of Lemma 3 and (33), we get that
Dividing both sides of inequality (36) by and letting , we obtain
On the other hand, substituting with , multiplying both sides of (33) by , and integrating it with respect to from to for , similar to (36)-(37), we get that
Dividing both sides of inequality (38) by and letting , we obtain
A combination of (37) and (39) yields
which contradicts (26). So the proof is complete.

Theorem 5. Under the conditions of Theorem 4, if for any sufficiently large ,
then (1) is oscillatory.

Proof. For any , let . In (41) we choose . Then there exists such that
In (42) we choose . Then there exists such that
Combining (43) and (44) we obtain (26). The conclusion thus comes from Theorem 4, and the proof is complete.

In Theorems 4 and 5, if we choose , where is a constant, then we obtain the following two corollaries.

Corollary 6. Under the conditions of Theorem 4, if for any sufficiently large , there exist with satisfying
then (1) is oscillatory.

Corollary 7. Under the conditions of Theorem 5, if for any sufficiently large ,
then (1) is oscillatory.

Theorem 8. Assume (5)–(7) hold, and there exists two functions and such that
where are defined as in Theorem 4. Then every solution of (1) is oscillatory or satisfies .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that on , where is sufficiently large. By Lemma 1 we have , where is sufficiently large and either on or . Now we assume that on . Let be defined as in Theorem 4. Then we obtain (32).Setting
Using Lemma 3 in (32) we get that
which is rewritten in the following form:
Substituting with in (50), an integration for (50) with respect to from to yields
which contradicts (47). So the proof is complete.

Theorem 9. Assume that (5)–(7) hold, and there exists a function such that , and has a nonpositive continuous partial derivative . If
where are defined as in Theorem 4, then every solution of (1) is oscillatory or satisfies .

Proof. Assume that (1) has a nonoscillatory solution on . Without loss of generality, we may assume that on , where is sufficiently large. By Lemma 1 we have , where is sufficiently large and either on or . Now we assume that on . Let be defined as in Theorem 4. By (50) we have
Substituting with in (53), multiplying both sides by , and then integrating with respect to from to yield
Then
So